Extremal mild solutions to Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions

In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a C-0-semigroup W(.) generated by the operator -A and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding C0-semigroup W(.) and non-instantaneous impulsive function ?(k) are compact, W(.) is not compact and ?(k) is compact, W(.) and ?(k) are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given.